Connections between singing and speech are indeed very deep. In Oliver Sacks' book, Musicophilia, he claims that humans alone among the animal kingdom possess an appreciation for rhythm. Consider for a moment the impacts of this. Can you have speech without rhythm? Well, no. Certainly rap artists would have a hard time. Did rhythm develop along with language, then, or before? Why and how did we develop into musical beings? Some early communication systems involved things like smoke signals, blasts of a ram's horn, or drumming. Is musicality a result of natural selection, perhaps for the sake of improved communication ability, which is an evolutionary advantage of humans?
Aniruddh Patel addresses these questions in his 2006 paper, Musical Rhythm, Linguistic Rhythm and Human Evolution. One of the first points to untangle is whether musical ability exists separately from speech, or whether they are entwined. A main argument against the notion that rhythm is an offshoot of language is that "beat perception and synchronization" is an aspect of musical enjoyment that has nothing to do with speech, which puts them at separate places at the cognitive table. Though the onset of language has been studied to death, there is little research about the onset of rhythmic perception. Why bother? Well, synchronization to a beat, for instance, could be crucial to the ability to segment syllables, a key to reading.
The interesting question arises, can animals learn to synchronize to a musical beat? If so, it would seem that it is not an evolutionary human adaptation. Here, Patel can only marvel that "despite decades of research in psychology and neuroscience in which animals have been trained to do elaborate tasks, there is not a single report of an animal being trained to tap, peck, or move in synchrony with an auditory beat."
Patel goes on to tie our linguistic ability to vocal learning, the need to develop the capacity to learn about the world from auditory codes. And it turns out this is not the unique purview of humans after all, but is shared by songbirds, parrots, and the whale family.
I personally find this move compelling, since I tend to resist any humans-only theories as being unduely influenced by our species' tendency for self aggrandization. The difference with humans isn't the adaptation itself, but the complexity of the resulting systems compared to others.
So sing to your kids. It will help them not only with language, but to learn many things from their auditory environment, possibly including over 1,000 birdcalls and the haunting, melancholy the song of the Humpback Whale.